Biologically-Plausible
Six-Legged Running:
Control and Simulation
by
Matthew David Malchano
Submitted to the Department of Electrical Engineering and Computer
Science
in partial fulfillment of the requirements for the degree of
Master of Engineering in Electrical Engineering and Computer Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2003
@ Matthew David Malchano, MMIII. All rights reserved.
The author hereby grants to MIT permission to reproduce and
distribute publicly paper and electronic copies of this thesis document
in whole or in part.
A uthor .........................
Department of Electrical Engineering and Computer Science
February 4, 2003
C ertified by ...............................................
Hugh M. Herr
Instructor, MIT/Harvard Division of Health, Science, and Technology
Thesis Supervisor
Accepted by..............
Arthur C. Smith
Chairman, Department Committee on Graduate Students
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
JUL 3 0 2003
LIRRARI.C
BARKFP
2
Six-Legged Running: Control and
Simulation
by
Matthew David Malchano
Biologically-Plausible
Submitted to the Department of Electrical Engineering and Computer Science
on February 4, 2003, in partial fulfillment of the
requirements for the degree of
Master of Engineering in Electrical Engineering and Computer Science
Abstract
This thesis presents a controller which produces a stable, dynamic 1.4 meter per second run in a simulated twelve degree of freedom six-legged robot. The algorithm is
relatively simple; it consists of only a few hand-tuned feedback loops and is defined
by a total of 13 parameters. The control utilizes no vestibular-type inputs to actively
control orientation. Evidence from perturbation, robustness, motion analysis, and
parameter sensitivity tests indicate a high degree of stability in the simulated gait.
The control approach generates a run with an aerial phase, utilizes force information
to signal aerial phase leg retraction, has a forward running velocity determined by a
single parameter, and couples stance and swing legs using angular momentum information. Both the hypotheses behind the control and the resulting gait are argued to
be plausible models of biological locomotion.
Thesis Supervisor: Hugh M. Herr
Title: Instructor, MIT/Harvard Division of Health, Science, and Technology
3
4
Acknowledgments
Hugh Herr, Gill Pratt, and Max Berniker, Joaquin Blaya, Bruce Deffenbaugh, Pete
Dilworth, Jon Gonda, Andreas Hofmann, Frank Kirchner, Marko Popovic, and
Rebecca Schulman:
Thanks much, y'all.
Mom, Dad, and Zach Malchano:
My love.
5
6
Contents
1 Introduction
13
2
Background Material
2.1 A Short Introduction to Running . . . . . . . . . . . . . . . . . . . .
2.2 Biological Inspiration . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Hexapedal Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
15
15
16
3
Hexapedal Running Control Algorithm
3.1 Introduction . . . . . . . . . . . . . . .
3.2 Morphology . . . . . . . . . . . . . . .
3.3 Running Control Algorithm . . . . . .
3.3.1 Aerial Phase Control . . . . . .
3.3.2 Stance Phase Control. . . . . .
3.4 Details of Methodology . . . . . . . . .
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Results
4.1 Motion Analysis . . . .
4.2 Perturbation Testing .
4.3 Robustness . . . . . .
4.4 Parameter Sensitivity .
4.5 Biological Plausibility.
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Conclusions and Future Work
45
A Mechanical Designs
47
B Control Source Code
B .1 6legs.h . . . . . . . . . .
B.2 create-hexahop.c . . . .
B .3 bug lhop.h . . . . . . . .
B .4 bug lhop.c . . . . . . . .
B.5 Miscellaneous Functions
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8
List of Figures
3-1
3-2
3-3
3-4
The hexapod model. . . . . . . . . . . . . . .
Diagram of hexapod model joint configuration,
Block diagram of aerial phase control . . . . .
Block diagram of stance control . . . . . . . .
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view
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from
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sagittal plane
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Mean kinematic motions of the center of mass, system potential and
kinetic energies, and pitch-axis angular momentum. . . . . . . . . . .
4-2 Mean ground reaction forces: horizontal and vertical components for
front, middle, and rear legs of a tripod. . . . . . . . . . . . . . . . .
4-3 Mean hip joint kinematics: angle, angular rate, and torque. . . . . .
4-4 Mean knee joint kinematics: leg length, velocity, and applied force.
4-5 Mean mechanical joint power output.
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4-6 Impulse torque perturbations: resulting momenta vs. time . . . . . .
4-7 Horizontal impulse force perturbation: resulting linear and angular
m om enta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-8 Vertical impulse force perturbation: resulting horizontal and vertical
linear m om enta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-9 Robustness to initial height and forward velocity . . . . . . . . . . . .
4-10 Robustness to initial pitch and pitch rate of change . . . . . . . . . .
19
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4-1
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A-1 Assembly drawings and section view of first distal segment design. . .
47
A-2 Assembly drawing and section view of revised distal segment . . . . .
A-3 Sectioned model of preliminary pneumatic distal segment . . . . . . .
49
49
9
10
List of Tables
3.1
3.2
3.3
Morphological parameters defining the simulated model. . . . . . . .
Aerial phase control parameters . . . . . . . . . . . . . . . . . . . . .
Stance controller parameters . . . . . . . . . . . . . . . . . . . . . . .
21
22
25
4.1
4.2
4.3
Summary of gait measurements. . . . . . . . . . . . . . . . . . . . . .
Maximum magnitude of step perturbations resulting in stable gait. .
Independent sensitivity of parameter values: minimum, nominal, and
maximum values for which a stable gait of more than 10.0 sec results.
Comparison of simulation measurements with biological model prediction s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
35
4.4
B.1 Variable names used in simulation source code and coresponding symbols from text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
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52
12
Chapter 1
Introduction
The difficulties of control in legged locomotion are well documented [28]: (1) the high
dimensionality and nonlinearity of the dynamics; (2) the relatively short duration
of and limited control provided by the ground reaction forces; (3) the complexity
of environmental conditions; (4) mechanical limitations in power, bandwidth, and
energy efficiency; and (5) operation in a gravitational field. Yet the promise of legged
locomotion is obvious in nature's ability to efficiently walk, run, trot, gallop, leap, and
bound over most of the Earth's terrain. The lack of machines capable of reproducing
this range of biological behaviors is evidence of the numerous challenges still faced by
researchers.
Many of the controllers for six-legged (or hexapod) robots exhibit a high level
of computational and conceptual complexity. Inverse kinematics, inverse dynamics,
and complex models are common features. The systems are defined by large sets
of high-precision parameters. The use of computational search methods, nonlinear
optimizations, and machine learning techniques is often necessary. The usual focus
is the production of a robust walking gait. Almost all of the resulting "hexapod
literature has [up until 2001] concerned static or quasi-static walking machines." [2]
Recent successes in legged locomotion have been reported using control techniques
labeled "feedforward", "open-loop", or "preflex" [34, 8, 33]. In these techniques, motions are coordinated by a central oscillator or clock. The oscillator triggers the
play-back of trajectories at each joint. Often the control targets a highly simplified
morphology with only one actuated degree of freedom (dof) per leg. The oscillator frequency, trajectory, and physical parameters are searched computationally or
empirically until a satisfactory form of locomotion emerges.
The algorithm presented in this thesis diverges from both the static walkers and
the "feedforward" approaches. In contrast to complicated control techniques used
in the walkers, the controller studied consists of only a few hand-tuned position and
velocity control loops and a two-state finite state machine. These loops servo joints
to simple positions or velocities. The hexapod model runs dynamically.
The controller resembles the "feedforward" methodology in its relative simplicity and its stabilization without the use of vestibular-type information, such as a
gyroscope signal. A number of important distinctions separate this work from the
"feedforward" class. The morphology, with two actuated degrees of freedom per leg,
13
lies between both the 3-dof insect leg design common in statically-stable walkers and
the 1-dof legs found in the "feedfoward" morphologies. The control is not a purely
clock-driven system, but rather it predicts approximate ground contact time using
force information in order to signal leg retraction. A single parameter value selects
from a range of forward velocities. Angular momentum information is used to couple
stance and swing legs motions. The controller uses the feedback of a few high-level
state variables and on-line calculation of dynamic quantities in generating leg trajectories. Finally, where "functional biomimesis" of cockroach locomotion inspires many
"feedforward" algorithms, the basis for this controller is instead a model of horse
trotting [15].
The organization of presentation here is as follows:
1. A short introduction of running locomotion, description of the horse trotting
model, and survey of hexapedal control is made in Chapter 2.
2. The main goal of this work is the design of a novel hexapedal running control
algorithm. It is described in depth in Chapter 3
3. The resulting motions of the gait produced in simulation, the evaluation of
stability through perturbation, sensitivity, and robustness testing, and an argument for biological plausibility may be found in Chapter 4
4. Finally, a summary of the main conclusions and future directions of this thesis
occurs in Chapter 5
5. I designed three mechanical leg segments to further physical testing, and they
are documented in Appendix A. The controller source code is listed in Appendix B.
14
Chapter 2
Background Material
2.1
A Short Introduction to Running
A run is a fast locomotory mode characterized by specific motions of the center of
mass. In particular, these motions feature significantly in-phase sinusoidal kinetic
and potential energies. The simplest model of running which exhibits these energy
profiles is the monopod. It consists of a single massless linear stiffness leg and a
point-mass body. It has been shown that many animals' legs approximate such linear
stiffnesses during running gaits [5]. Additionally, many runs feature an aerial phase:
a period during which no feet contact the ground [24]. These characteristics hold true
for morphologies running with any number of legs [4, 10].
A common gait used by hexapods, both natural and artificial, is the alternating
tripod gait [17]. This symmetric and alternating gait moves legs together in groups of
three. The front and hind ipsilateral and middle contralateral legs combine to form a
tripod. The legs of a given tripod alternate between stance phase (during which they
contact the ground) and swing phase (during which they move forward through the
air).
Gaits, including running gaits, may be divided into those that are statically stable
and those that are dynamically stable. This terminology was introduced by [23]. The
points of ground contact of the stance legs define a support polygon. In a statically
stable gait, the center of mass always lies directly above this polygon. In a dynamically
stable gait, on the other hand, center of mass projects outside of this polygon for some
significant portion of the gait cycle. The gait then relies on its momentum to carry
it through these periods of instability. Cockroaches, even though six-legged, move
dynamically [12].
Given these above definitions, the hexapod presented in this thesis may be said
to move in a dynamically-stable, alternating-tripod running gait.
2.2
Biological Inspiration
A biological model of horse trotting [15] provides the initial basis for the running
controller presented in this work. This quadrupedal model has been shown to suc15
cessfully predict biologically observed metrics across a range of masses [14], and has
been further adapted to model horse galloping gaits [16].
Trotting is a gait where diagonal limb pairs alternately strike the ground. This
pairing resembles the motions of a hexapedal alternating tripod gait, where limbs are
also grouped diagonally. Trotting occurs as a middle speed gait between walking and
galloping in a variety of quadrupedal animals. The central ideas posed in the trotting
model are:
1. Both diagonal limbs in a pair begin retraction simultaneously.
2. A trotting animal estimates when its feet will first strike the ground and begins
limb retraction at this time. Hence, retraction may occur before foot contact.
This estimation is performed on-line using the previous stance phase pitch and
force information. It is believed that beginning retraction before ground contact
may help stabilize those gaits which may be modeled as monopods [35], such
as trotting and hexapedal running.
3. Stance leg retraction attempts to match a tangential target speed.
4. Vertebrate limbs behave like linear stiffness springs during ground contact.
5. Hind hip joints provide motive force. Forward hip joints provide braking force.
6. No active control of body orientation is used.
The trotting study was conducted using sagittal planar models. These models
possessed 4 radially compliant legs with rotary hip joints; back and neck joints with
passive stiffnesses were included to replicate the horse morphology accurately. Stability was evaluated based on the number of gait cycles the simulation locomoted
without significant change to pitch, aerial apex height, or forward velocity. Additionally, simulations were subjected to a ground impedance change while trotting to test
disturbance rejection.
The above ideas provide both the inspiration and starting framework for this work.
Specific documentation of divergences and modifications may be found in Chapter 3.
2.3
Hexapedal Robots
This section seeks to place this thesis' contribution in the context of other work in
hexapedal control. The histories of the study of legged locomotion are long and
broad, and more detailed accounts of the robotics [31], biomechanics [24], and recent
advances [32] may be found elsewhere.
The literature documents numerous approaches specific to hexapedal control.
These approaches have focused mainly on producing robust walking gaits in 3-dof
per leg morphologies. The advanced concepts and computational methods are applied. Parameter search techniques have been used to reduce the complexity of the
resulting controllers. For example, neural nets [3], genetic algorithms [22], and varieties of reinforcement learning [18] have been used to define parameters. The problem
16
has been decomposed into a variety of subproblems, with restricted goals such as gait
generation [39], fault tolerance [41], rough terrain navigation [40], and leg coordination [1]. Biological organisms have served as physical [38, 26] as well as behavioral
models [11]. Many of these controllers are successful in their chosen objectives or
introduce innovative approaches. Yet all of these hexapods, along with many others, locomote in only static or quasi-static regimes. Their focus includes neither fast
running gaits nor simplicity.
The MIT Leg Lab has previously developed two six-legged walking simulations
similar to the aforementioned systems. The first controller targets a cockroach-sized
morphology and uses a technique called "virtual model control" [37]. In virtual model
control, desired forces are prescribed to the body link. For example, a "virtual"
linear spring, attached between the center of mass and a "virtual" fixed frame, might
counteract gravity's force on the body. The leg inverse kinematics and the Jacobian
are used to map these "virtual" forces to leg joint torques, which exert real forces
against the ground to produce the desired motion. The second hexapod, a simulated
automobile-sized walking vehicle, also relies on this technique [27]. Neither controller
claims biological plausibility.
More recently, impressive successes have been had in a series of highly simplified
approaches. These approaches are unified in their use of a clock to coordinate the
playback trajectories on 1-dof actuated legs and their emergent passive stability. They
are also unified by their "functional biomimesis" of cockroach locomotion. Below I
discuss RHex [2, 34], Sprawlita [8, 6], and Wheg [30], three examples of this approach
to control.
The autonomous robot RHex possesses six actively controlled degrees of freedom,
one per leg. The actuation is provided by rotary hip joints whose axes lie perpendicular to the sagittal plane. Legs move in complete rotations and group together in
alternating tripods. A central oscillator coordinates the hips, which use local feedback loops to follow one of two velocities: a "retraction" velocity for stance phase
and a faster "protraction" velocity for swing phase. Buhler et al. report a forward
gait, with a speed of one body length per second, emerges after physical tuning of
the system mass, leg spring stiffness, target velocities, and loop gains.
Sprawlita is a pneumatically actuated, tethered robot. It's 1-dof legs are pneumatic cylinders attached to the body with passive stiffnesses. In Sprawlita, a central
pattern generator again sends a periodic feedforward trajectory to the mechanical
system. Cham et al. argue passive stiffnesses and pneumatic cylinders act as purely
mechanical feedback (termed "preflexes") which stabilize the gait. Otherwise, no
feedback occurs. The controller may also servo the angle of all hips, but does not do
so during intra-stride operation. The feedforward trajectory is a simple "bang-bang",
extend-or-retract command sent to all legs in a tripod. The frequency and duty cycle
of this trajectory were determined empirically. Sprawlita has been shown to move at
speeds of 2.5 body lengths per second.
Wheg uses a morphology similar to RHex but with three radial-stiffness spokes
per leg instead of one. Additionally, it reduces leg drive to a single motor. Its
controller appears similar to RHex, but targets only a single angular rate per leg and
switches its legs to an in-phase gait when climbing obstacles. It's forward velocity is
17
undocumented.
Since the initial work on both Sprawlita and RHex, additional research has begun
adding high-level adaptive feedback to these systems. In RHex, body orientation and
foot contact information has been used to adjust for ground slope conditions [20].
This information is also being used to research adding an aerial phase to RHex's
gait. In the above, Komsuoglu et al. state that, "experimental observations suggest
higher forward speeds can be achieved provided an aerial phase is introduced....". In
Sprawlita, a foot contact signal was used to modify the frequency and duty cycle of
the feedforward trajectory to aid slop ascent [7].
Both of these recent incremental successes, I believe, validate the direction of the
work pursued in this thesis, which utilizes a different set of high-level feedback terms
in a simple control system.
18
Chapter 3
Hexapedal Running Control
Algorithm
3.1
Introduction
Figure 3-1: The hexapod model.
The control algorithm presented in this chapter generates leg motions in the above
12-dof model (Figure 3-1). These leg motions in turn produce a fast tripod running
gait in simulation.
What are the salient features of this control? It is coordinated by a finite state
machine. Foot contact switches signal FSM state changes. Proportional gain or
proportional-derivative feedback loops mostly servo joints to constant positions or
velocities. No active mechanism maintains body orientation; instead, level posture
19
emerges passively. Leg retraction matches the distal leg end velocity to the target
forward velocity. Angular momentum information about the stance leg is used to
coordinate opposing swing leg motion. Leg retraction occurs based on an on-line
prediction of ground contact time. The resulting gait possesses an aerial phase. The
rest of this chapter gives precise description of these features.
Simulations of sagittal-planar and fully three-dimensional models were developed
for a variety of masses. The results contained in this thesis focus only on the sagittalplanar model. The hexapedal sprawled posture causes most of the interesting dynamics to occur in the sagittal-plane. Reduction of orientation state to only pitch
angle and derivatives allows a clear examination of the degree of passive orientation
stability. For the three-dimensional simulations, only small changes to leg sprawl and
increases in hip gains were necessary to achieve stable locomotion.
Section 3.2 describes the model used in simulation. The control algorithm is discussed in detail in Section 3.3. As previously mentioned, this work draws inspiration
from a model of quadrupedal trotting. Documentation of the specific differences between these systems, along with various notes on simulation development, may be
found in Section 3.4.
Results analyzing the gait produced by this control follow in Chapter 4. The
complete source code for the controller, along with some supporting simulation code,
may be found in Appendix B.
3.2
Morphology
The hexapod model possesses 12 actuated degrees of freedom. Each leg has a rotary
hip whose axis lies perpendicular to the sagittal plane and a prismatic knee oriented
radially from that hip. The hip joints attach the thigh link to the body, with fore and
hind hips located at the body's corners and at the vertical level of the body center of
mass. Each hip joint acts to protract (swing forward) and retract (swing hindward)
a given leg in the sagittal plane. The knee joints attach the thigh to the shin and act
to lengthen or shorten the leg. Additionally, during stance phase, these knee joints
behave as linear, high stiffness leg springs. Figure 3-2 diagrams these actuated joint
motions in the sagittal plane.
An un-actuated joint, co-located with the hip, splays the thighs outward within
the frontal plane. A passive 40 N/in stiffness and 20 N-s/in viscosity hold these legs at
an angle of 0.1 radians from vertical. The splay and stiffness maintain the necessarily
sprawled posture in the 3D simulations.
The hexapod inherits this general form from the quadrupedal trotting model.
Similar morphologies also appear in other robots and biomechanical models, such as
[31, 25, 36], in part due to common recognition of a tested hypothesis suggesting that
the legs of many animals behave like linear-stiffness springs [5].
The physical parameters defining the hexapod are summarized in Table 3.1. The
total mass, its distribution, body dimensions, and leg lengths roughly approximate
the latest iteration of the Scorpion robot [19]. I chose to model these values on the
Scorpion because of prospective plans to conduct physical tests using some of its
20
6A
Figure 3-2: Diagram of hexapod model joint configuration, view from sagittal plane
hardware. One consequence of this choice, the legs total nearly 50% of the system
mass and hence contribute significantly to the system dynamics.
Parameter
Symbol
Value
Unit
Body mass
-
3.104
kg
Thigh mass
Shin mass
Mthigh
Mshin
0.464
0.029
kg
kg
6.062
kg
Total mass
Body length
-
0.64
m
Body height
Body width
Thigh length
Thigh radius
Shin length
-
m
m
m
m
m
Lshin
0.10
0.145
0.076
0.15
0.176
Shin radius
-
0.024
m
Max Leg Length
-
0.252
m
Min Leg Length
Max Hip Angle
-
0.052
m
rad
Min Hip Angle
-
-
Lthg9 h
-
7r/2
-7r/2
rad
Table 3.1: Morphological parameters defining the simulated model.
21
3.3
Running Control Algorithm
This section discusses the control algorithm which generates the hexapod's running
gait. A centralized controller coordinates the motions of the legs. This controller is
in one of two states during steady state locomotion: aerial phase (Section 3.3.1) or
stance phase (Section 3.3.2). Each state performs a separate function. In the aerial
phase, the hexapod moves in a ballistic trajectory through the air. The controller
positions the legs to prepare for ground contact. During the stance state, the hexapod
contacts the ground, and the controller moves the legs in order to propel the body
forward and upward into the air.
In both states, the legs are grouped into two tripods of three legs each. Each leg
in a tripod is commanded similarly. At any time, one of the tripods is designated
the stance tripod and the other the swing tripod. During stance phase, the legs of
the stance tripod are in contact with the ground, while the legs of the swing tripod
do not touch the ground. When control switches from the stance phase to the aerial
phase, the designations for the tripods are swapped. The swing tripod becomes the
stance tripod and vice versa. Hence, in aerial phase, the tripod designated as the
stance tripod will contact the ground in the subsequent stance phase.
3.3.1
Aerial Phase Control
Aerial phase control positions the legs to prepare the hexapod to enter into stance
phase control. Aerial phase control begins when the hexapod's leg leave the ground
and ends when ground contact is predicted to occur again. During aerial phase control, the legs are positioned to a ready position for the next stance phase's initiation.
Figure 3-3 shows the aerial phase controller for both knee and hip joints in block
diagram notation. Table 3.2 lists the numerical values of the controller gains and
target positions.
Parameter
Kaerial knee
Baerial knee
Kaerial hip
Baerial hip
Lknee extend
Lknee retract
Ohip protract
khip retract
Value
2.0
1.086
40
Unit
N/m
Ns/m
N/rad
Description
Aerial knee stiffness
Aerial knee damping
Aerial hip stiffness
20
0.0
0.2
Ns/rad
Aerial hip damping
m
m
rad
rad
knee target compression
knee target compression
hip target position
hip target position
-0.4
0.4
Table 3.2: Aerial phase control parameters
At the beginning of the aerial phase, the swing tripod starts in a retracted position
and with its knees fully extended. The stance tripod starts protracted and with its
knees fully compressed. Proportional-derivative position control loops at the hips
22
apply torques to both tripods to brake the inertia from the previous stance phase's
motions and to roughly position the tripods for entry into the next stance phase.
The target angle for the stance tripod protraction was estimated by considering the
return path of a spring-mass monopod dropped from the predicted average height
of the hexapod center of mass apex and with the predicted velocity of the hexapod
robot. The angle of attack of this monopod was used as the target protraction angle.
This value was then hand-tuned until a steady state run in the hexapod model was
achieved. The same angle from vertical is used for the swing tripod target retracted
position.
Hip loop gains were chosen to assure that legs were brought to a halt before they
contacted the hip joint stops. Although it was intended that loop gains would be
high enough to assure precise positioning to the desired target positions before the
end of aerial phase, this behavior was not observed in the final simulation. Instead,
some variation in protracted leg position was observed. Hip damping coefficients
were chosen to critically damp the motion of the hips, assuring that no oscillation or
overshoot occurs during leg motion.
When aerial phase is entered, the controller estimates the time of the next ground
contact, which is when stance phase should start. This estimation is accomplished
using the vertical velocity of the center of mass of the hexapod at the moment aerial
phase begins. From this, the duration of a parabolic flight path during aerial phase is
predicted. A timer maintains the elapsed time from the onset of aerial phase control;
when the timer exceeds the predicted duration, aerial phase ends and stance phase
begins. The controller also switches into stance phase if a leg contacts the ground at
any time during aerial phase motion.
Just before aerial phase control ends, P-D controllers apply forces to the knees
to extend the stance tripod to its maximal length and compress the swing tripod by
0.15 meters. Knee gains and damping coefficients were chosen to assure this motion
completes before ground contact.
23
S 0.4
Protract Angle
[rad]
+_a[OUTPUT]
Actuator
Model
+_K
[STATE]
STANCE/SWING
Desired Hip Angle
-m 4
Retract Angle
[rad]
K aerial hip
[rad]
Applied Hip
Torque
[SENSOR]
Hip Position
Derivative
Baerialhip
[Nm
[m4lrdF
0.4
Extend Positon
[m]
STAE]N
+_pie
[rad]
STNC/WIG
Desire Ke Plos
KK+n[UTUT
K aerial knee
Actuator
Mde
[m]
-0.4
Compress Position
[m]
|[SENSORI
Knee rPosition
ud
Derivative
B
aerial
knee
Figure 3-3: Block diagram of aerial phase control
24
Applie KNe
Force [N]
3.3.2
Stance Phase Control
Stance phase control determines the motions of the tripods during ground contact.
The stance tripod usually contacts the ground shortly after the onset of stance phase
control, and the control ends when that tripod again leaves the ground. During this
control, the stance tripod moves from a protracted position to a retracted position
and acts against the ground. Simultaneously, the swing tripod moves from a retracted
position to a protracted position without contacting the ground. Figure 3-4 shows a
control block diagram depiction of the control used during stance phase. Table 3.3
lists the gains and desired velocities used in the controller.
Parameter
Gvel
Gve12
Value
6.0
-0.2
Xmax
1
compress
Fore/Hind kleg
Middle kie,
bleg
1.5
0.22
750
1500
20
Unit
N-sec/rad
N-sec/rad
m/sec
m
N/m
N/m
N-sec/m
Description
stance tripod loop gain
swing tripod loop gain
target forward velocity
swing leg compressed length
stance leg stiffness
stance leg stiffness
stance leg damping
Table 3.3: Stance controller parameters
Stance phase control begins when either the predicted duration of the aerial phase
expires or when one or more legs contact the ground. In normal steady-state behavior,
stance phase initiates slightly before ground contact. This preliminary acceleration
results in "leg retraction", which was one of the hypotheses posed in the quadruped
model.
At the initiation of stance phase control, knees begin to act as high linear stiffnesses. The front and hind stance legs are set to a 750 N/m stiffness with 20 N-sec/m
damping coefficient. The middle leg switches to a 1500 N/m stiffness; this stiffness is
doubled to compensate for the asymmetry of the tripod. Without this adjustment,
asymmetrical ground reaction forces, a body-roll moment, and insufficient rear leg
ground clearance occur during stance. Swing tripod knee stiffness also increase.
The hips of the stance tripod are controlled with proportional velocity loops.
These loops attempt to match the distal end of a stance leg's tangential speed to the
ground speed for speeds less than the desired forward velocity imax. The distal end
of the leg has a velocity of v = Ileg * #hip where lie,(t) is the length of the leg and
#hip(t) is the angular rate of the hip joint. The torque Thip(t) applied is:
Thip(t) = Gvei * ((lieg(t)
-
hip(t)) -
Ximax)
(3.1)
The swing leg control operates as follows. Proprioception information provides
the compression of the stance leg Alstance(t), along with the velocity of the stance leg
hip #stance(t) and of laterally-opposed swing leg hip #swing(t). The moment of inertia
25
for the stance leg about the hip joint is calculated:
Istance
I Mthigh -
(0.5 - Lthigh) 2 +
Mshin - (0.5Lihin
- Lthigh -
Alstance(t)) 2
(3.2)
Hence, the angular momentum of the leg about the hip axis is
Lh p(t) = 'eg(lieg(t))
The target velocity
bswing(t)
(3.3)
- Oip.
of the stance leg is set to cancel this angular momentum.
Istance (t)
=
-
(3.4)
Iswing(t) -/Ostance(t)
A proportional velocity control loop with negative gain
speed to follow bswing(t). The applied hip torque is
Thip(t) = Gvei 2 - (bswing(t) +
Gel2
swing (t)).
servos the swing leg
(3.5)
Since the swing leg length is always shorter than the stance leg length, Ising(t) <
Istance(t) and therefore iswing(t) > &ance(t).
The swing leg will hence be in position
to enter aerial phase. We suspect that angular momentum about the center of mass
plays an important role in dynamically stable locomotion. Swing leg control annuls
the pitch-axis angular momentum about the hip, but this is only the smaller of the
two terms expressing the effects of a leg's contribution to the angular momentum
in the body frame. Further research is necessary before these speculations can be
confirmed.
The control outlined above for the hip and knee joints is maintained throughout
stance phase. Stance phase ends when all feet leave the ground. At the transition,
the controller starts the aerial phase timer, records the vertical velocity, and switches
stance and swing tripod designations. It then enters aerial phase.
26
xdot-max
d/dt x desired
[m/sec]
!:
Stance Leg
Length [m]
[SENSOR]
+
G-vel
[OUTPUT]
Actuator Model
-x
Stance Hip
Gain
Product
Stance Hip
Torque
Nm
ESENSOR]
Stance Hip
Angular Velocit ty
[rad/sec]
Mx
Swing Leg
Length [in]
x
Stance Leg
Moment of Inertia
G_ve12
[PTPT
Product
Actuator
Product.
Swing Hip Gain
Mdl
Swing Hip
Torque
[N-m]
[(SEN SOR]
Swing Leg
Moment of Intertia
Invert
[SE NSOR
Swing Hip
Angular Veloci ty
[rad/sec]
0.25
Leg Ext
Length [m]
K-leg-,
[OUTPUT]
Stance Leg StiffnessJ[N/m]
Derivative
Actuator
Model
Stance Knee
Force
[N]
Actuator
Model.
Swing Knee
Stance Leg Damping
[Ns/m]
[0.22
Leg Comp
Length [m]
+~
Kswg
-,
Swing Leg Stiffness
[N/m]
d/t
B-swg
Derivative
Swing Leg Damping
[Ns/m]
Figure 3-4: Block diagram of stance control
27
[OUPUT]
Force
[N]
3.4
Details of Methodology
The control presented here uses the quadrupedal model as inspiration, but both
morphology and control differ in a number of very significant ways. The hexapod
morphology adds two legs, makes the body link rigid, and removes the head link. The
mass, mass distribution, and dimensions have all been changed. Viscous damping has
been added to the hexapod's stance leg springs to model mechanical losses.
Control differs from the quadruped as follows. Symmetric target velocities and
stiffnesses are used for fore- and hindlimbs; in the quadruped these targets are significantly different. The aerial state occurs when all feet leave the ground; in the
quadruped it occurs when any single foot loses contact. Only vertical take-off speed
is measured to predict ground contact; the quadruped also uses pitch sensor information. The swing leg follows the opposing stance leg's angular momentum; the
quadruped's swing leg "mirrors" the position of the stance leg instead. Parameter
values differ widely. The quadrupedal model documents that twenty-eight unique
velocities, stiffnesses, gains, and feedback targets define its control. The hexapod
control is defined by thirteen such quantities.
Simulations were developed in Creature Library [21], an MIT Leg Lab extension to
the commercial dynamics simulator SD-Fast [9]. A 4th-order Runge-Kutta integrator
integrated the equations of motion generated by SD-Fast with a 200 nsec step size.
The ground model was a linear spring and damper system with a stiffness of 400 N/in
and a viscosity of 100 N-sec/m.
All simulation results reported relied on hand-tuned control parameters. At one
point, I attempted to computationally generate parameters using naive nonlinear
optimization. The optimization attempted to minimize the basic cost function
V
=
1
(-i(t) - Xd)
2
+ (z(t) - Zd)2 +
(t) 2
(3.6)
to<t<tf
with V evaluated using the simulation. In the case of instability, out of range parameters, or other failure cases, V defaulted to its maximal value. The search space was
over only five controller parameters. Computing an analytical gradient for the system
appeared too difficult an undertaking, and hence Powell's method [29] was chosen as
the nonlinear optimizer. It generates gradient information by sampling cost function
values for changes in each parameter. It then line-minimizes along the direction which
appears most promising. Initial parameter values were selected from a known-good
starting point.
This approach was proved infeasible due to a number of stumbling blocks. The
cost function failed to capture the desired behavior. The optimization seemed to
become trapped in local minima. Large portions of the parameter space produce
no useful optimization information due to the "brick wall" nonlinearities in the cost
function for gait failures. The simulation runs far slower than realtime and an evaluation of a parameter set requires on the order of tens of seconds of simulated time.
Hence optimization trials took a long time. Numerical optimization was eventually
abandoned in favor of hand-tuning.
28
Chapter 4
Results
A mathematically rigorous, analytical evaluation of system stability is not tractable:
the high dimensionality, nonlinearity, and dynamic complexity alone limit the ability
to present such a treatment. Even if it were possible, the the value of the analyses
obtained would be questionable. Yet, the existence of a stable gait in simulation provides thin proof unless its ability to handle the wide range of environmental conditions
found in the real-world is demonstrated.
This chapter presents evidence that argues for the system's convergence to a
steady-state periodic gait over a range of initial conditions, in the presence of disturbances, and with an insensitivity to changes in parameter values. These tests offer a
compelling argument that the algorithm exhibits long-term stability and is physically
realizable.
Simple metrics of the resulting gait (Table 4.1) allow performance comparison to
other legged locomotion algorithms. Mean kinematic traces for joints and the center
of mass motions describe periodic steady-state behavior (Section 4.1) . Error bars
on these kinematic traces provide information as to the typical divergence seen from
this mean motion. Torque and force perturbations quantify disturbance rejection
(Section 4.2). Robustness testing samples the space of initial conditions to determine those which result in steady-state motion (Section 4.3). Parameter sensitivities
measure the necessary precision of the values defining the controller (Section 4.4).
Finally, the biological plausibility of the algorithm is discussed (Section 4.5).
Throughout this chapter, a gait is termed stable or in a periodic steady state if,
over some stated period of time, the gait occurs without a gait failure. A gait failure
is defined as body ground contact or joint torques exceeding simulation integration
limits. The system is considered stable if it exhibits a stable gait. Additionally, when
initial conditions of a simulation are not given, it may be safely assumed they are
standard initial conditions: J = 0.7 m/s; z = 0.3 m; 0 = 0.0 rad; 0 = 0.0 rad/sec;
and test
contact =
0.09 sec.
29
Parameter
Mean Velocity
Stride Length
Stride Frequency
Value
1.48
2.35
0.71
2.08
Units
m/sec
body lengths/sec
meters
Hz
Tripod Duty Cycle
30.2
%
Mean Mech Power
Peak Mech Power
38.88
481.0
Watts
Watts
Froude Number
0.957
-
Table 4.1: Summary of gait measurements.
4.1
Motion Analysis
The gait analysis plots below display the mean motions of the hexapod during steadystate locomotion over a single gait cycle. This gait cycle includes the stance phases
of both tripods along with two aerial phases and lasts an average duration of 0.435
sec. The stance phase for one tripod begins at t = 0.00 sec and ends at t = 0.13 sec;
the opposing tripod's stance phase begins at t = 0.21 sec and ends at t = 0.34 sec.
Figure 4-1 shows the motions of the center of mass along with the system's kinetic
and potential energy and angular momentum in the center of mass frame.
The remaining plots give leg specific information for the tripod initially entering
stance. Since the gait is nearly symmetric, only a single tripod's motions needs
be displayed. Figure 4-2 shows the ground reaction forces for the front, middle,
and rear legs of this tripod. These forces are split into vertical (z-direction) and
horizontal (x-direction) components. Positive horizontal forces point in the direction
of robot motion. Figure 4-3 shows the hip joint position, velocity, and force for the
same legs. Positive values correspond to retraction. Figure 4-4 shows the knee joint
position, velocity, and force. Figure 4-5 shows absolute mechanical power output for
the individual joints. These power graphs do not account for regeneration through
negative work. They also neglect possible passive mechanical implementations. For
example, if the knee joint stiffness in stance phase were implemented with a passive
compression spring, no net power output would occur. Previous power numbers do
take passive knee implementation into account.
These plots were produced in a standard gait plot format. The data were collected
over a 10 second simulation. They were then split into samples beginning at the
instant of a specific tripod's first ground contact. These samples were resampled
to a uniform length and filtered with a 10-point window, low-pass FIR filter. The
mean across these samples was then taken and is plotted as the solid line. The two
dashed lines delimit plus and minus one standard deviation from this mean. Some
minor high-frequency noise is visible at the edges of a few of the plots. This noise
is believed to be an artifact of the FIR filter; it should be stressed that it does not
result from any system motion or dynamics.
30
Body Pitch
COM Horz. Velocity
COM Height
-0.025
1.6
0.24
-1.5
E
0.22
'0
E
-0.03
1.4
4)1.3
0.2
0.04
1.2
0.18
-0.0 5
1.1
0
0.2
0
0.4
0.2
0
0.4
Ang Mom Y
Kin. Energy
Pot. Energy
14
0.1
9.5
0.08
0.06
9.
13
8.5
12..
0.04
0.02
8
11
7.5
0
71
10
0
0.2
6.5
0
0.4
0.4
0.2
-0.02
-0.04
-...
0
0.4
0.2
0.2
0.4
Figure 4-1: Mean kinematic motions of the center of mass, system potential and
kinetic energies, and pitch-axis angular momentum.
GRF Hind Leg (x)
GRF Middle Leg (x)
GRF Front Leg (x)
5
5
Z
0
0
z
2.
z
0
-5
0.
-5
-
-10
0
0.2
0.4
0
0.2
0
0.4
40
30
0.4
GRF Hind Leg (z)
GRF Middle Leg (z)
GRF Front Leg (z)
0.2
70-
50
60
40
50
z
20
30
40
z3
z
20
10
20
10
10
0
0
0
0
0.2
0.4
0
0.2
0.4
0
0.2
0.4
Figure 4-2: Mean ground reaction forces: horizontal and vertical components for
front, middle, and rear legs of a tripod.
31
Hind Hip Pos
Mid Hip Pos
Front Hip Pos
0.5
0.5
0'
0
01
0.5
--
0
0.2
-0.5
-0.5
0.4
0.2
Mid Hip Vel
0
Front Hip Vel
S5
0.4
0
5
5
0.
01
0.2
Hind Hip Vel
0.4
0.2
Hind Hip Tor
0.4
0.
-5
-5
0
0.2
Front Hip Tor
-5
0.2
Mid Hip Tor
0
0.4
0.4
0
4
E
3
3.
2
2
0
0.2
2.
0
02
0
-
0.4
0
00.2
0.4
0
0.2
0.4
Figure 4-3: Mean hip joint kinematics: angle, angular rate, and torque.
32
Hind Knee Len
Mid Knee Len
Front Knee Len
0.25
0.25
0.25
0.2
0.2
0.2
0.15
0.15-
0.15
-_
0
0.4
0.2
0
0.4
0.2
Front Knee Vel
4
2
2
0
0-
0
0.4
0.2
Mid Knee Force
0
Front Knee Force
30
0.4
-4
-4
0.4
0.2
0.2
Hind Knee Vel
-2
-2.
0
0
Mid Knee Vel
2
LL
.
0.4
0.2
Hind Knee Force
40
60
20
40
20
0
20
-10
0
0
0
-_________
0.4
0.2
0.2
0
0.4
0.2
0
0.4
Figure 4-4: Mean knee joint kinematics: leg length, velocity, and applied force.
Hind Hip Power
Middle Hip Power
Front Hip Power
151
1
15
10
-
-0
100
800
0.4
0.2
0
20
5
Front Knee Power
00
0
0
0.2
0.4
0.2
04
Hind Knee Power
Mid Knee Power
400
40
0.2
0
0.4
0
0.2
0.4
0
0.2
0.4
Figure 4-5 Mean mechanical joint power output.
33
4.2
Perturbation Testing
Perturbation testing measures the influence of environmental disturbances on the
stability of a controlled system. In the case of the hexapod, torques and forces are
applied to the center of mass during stable locomotion. The resulting linear and
angular momentum response is examined. Impulse and step function perturbations
are employed to measure the rejection of near-instantaneous and continuous changes
to these momenta.
Many detrimental environmental conditions, to a first approximation, might be
modeled as unforeseen, sudden changes to system momenta. For example, impacts
with small obstacles, abrupt ground height variations, or foot contact slips alter the
hexapod's linear and angular momenta. Uneven ground impedance or hip torque
mismatch change the angular momentum. Results demonstrating the return of momenta to pre-perturbation levels, along with upper bounds on rejected disturbance
magnitudes, provide evidence that stable gait survives environmental disruption.
Short duration (0.8 msec) torques were applied directly to the center of mass.
These approximate a zero time duration impulse but, unlike a zero time duration
force, cause a non-zero change in system angular momentum. Perturbations were
applied at t = 5.1 sec, which corresponds to the instant of a stance phase initiation.
Figure 4-6 shows the magnitudes of the pitch-axis angular momentum (top row)
and x-axis linear momentum (bottom row) over two separate simulation runs. A
dashed line delineates the instant of perturbation. In the left plot of this figure, a
positive torque "impulse" of magnitude 75 N-m acts on the center of mass, producing
a forward pitching motion. In the right plot, a negative torque of magnitude -100
N-m is applied to the center of mass again at t = 5.1 sec.
In both cases, the disturbance induces a sudden visible pitching of the hexapod
which continues as an oscillation for a number of gait cycles. Eventually, this oscillation dies off and a stable gait continues. As the plots indicate, pre-perturbation angular momentum amplitude initially lie almost entirely within a -0.5 to 0.25 kg -m 2 /sec
range. At the time of impulse, it peaks sharply to a magnitude of nearly 0.65
kg - m 2 /sec. Shortly thereafter, it returns to its previous range.
Similar to the use of torques to test dynamic rejection of sudden angular momentum changes, short-duration forces were used to study the effects of sudden changes
in linear momentum. These forces acted on the center of mass, again at t = 5.1
sec, again for a duration of 0.8 msec , and in each of four directions (upward, downward, forward, and hindward). Figure 4-7 shows the angular momentum (top row)
and x-axis linear momentum (bottom row) effects originating from horizontal force
perturbations. Figure 4-8 shows the x- and z-axis linear momentum response to
vertically-directed forces.
A different class of perturbation tests were then performed. These tests measure
the ability of the controller to reject continually applied, or "step" disturbances. At
t = 5.1 sec, a force or torque is applied to the center of mass for the remainder of
the simulation. If the hexapod completes the 10.0 seconds following perturbation
without a gait failure, the magnitude of the perturbation is increased. This process is
repeated until instability occurs after the step disturbance begins. In this manner, a
34
Angular M~tu
(pitch)
0.6
0.6
0.4
0.2
0
02-
_0.2
-0.4
_0A
4
5
6
9
7
10
11
5
12
a
7
9
10
11
12
10
11
12
Ur*& MOrantUmn (X)
Ulnear Momnantumn (x)
9.5
9
-86.5
7.5
7
6.5
H7
4
5
6
7
a
9
10
11
6
5
12
7
a
9
(b) -100 N-m at t=5.1sec, 0.8 msec
(a) 75 N-m at t=5.1sec, 0.8 msec
Figure 4-6: Impulse torque perturbations: resulting momenta vs. time for (a.) positive and (b.) negative torques.
loose upper bound on rejected step magnitudes is established to at least two significant
figures. Table 4.2 lists these maximal step magnitudes.
Perturbation
Torque Step
Force Step
Units
N-m
N-m
N
Direction
positive
negative
forward
Max Mag
4.1
2.75
17.9
hindward
5.5
N
upward
downward
21.75
8.1
N
N
Table 4.2: Maximum magnitude of step perturbations resulting in stable gait, for at
least 10.0 sec after perturbation. Perturbations applied for all time after 5.1 sec.
The angular momentum of the hexapod during normal locomotion is relatively
low compared with the bounds of disturbance generated momentum peaks. Recent
research within the lab has produced preliminary evidence that human walking and
other locomotive behavior possesses a near-zero instantaneous system-wide angular
momentum magnitude. This may relate to the overall stability occurring in the model.
Further investigation is necessary.
What physical conditions can these step perturbations be seen to model? Continual forward and hindward forces occur during hill climbing and could be seen as
velocity errors and certain ground conditions. Downward and upward forces also occur on slopes, test resilience to controller ground contact time mis-predictions, and
evaluate operation in other gravitational fields. It is more difficult to reasonably
interpret step torques; they should be considered as merely quantifying maximum
35
Angular Morentum (tch)
Angular Mmentum (ptch)
0
0.6
.2-
0.4
0.1
0.2
0
0.1
-0.2
5
7
6
8
9
10
5
11
6
unear
uear Momentrn (x)
12
I
9
13
-
6
8
7
7
1
9
10
11
12
10
11
12
Momrentum(x)
I-
.
8
11
'7
'10
6
9
8
5
6
8
5
L9
10
5
11
6
7
8
9
(b) Rearward 600N, at t=5.1 sec, 0.8 msec
(a) Forward 600N, at t=5.1 sec, 0.8 msec
Figure 4-7: Horizontal impulse force perturbation: resulting linear and angular momenta.
rejected angular momentum error for lack of a natural cause.
36
Linear Momentum (z)
Linear Momentum (z)
4 -
2
-2
0
-2-4
-6 -*
-4
S
6
7
8
9
Uea Mmu
10
11
5
12
11
6170
x
I
5
. [
7
5
6
7
a
9
10
5
i1
a7
10
11
(b) Downward 400N, at t=5.1 sec, 0.8
msec
(a) Upward 200N, at t=5.1 sec, 0.8 msec
Figure 4-8: Vertical impulse force perturbation: resulting horizontal and vertical
linear momenta.
37
4.3
Robustness
To test the robustness of the controller, a subspace of initial dynamic conditions is
sampled. The goal is to determine the area inside this subspace for which stable,
long term locomotion results. If locomotion is maintained for some defined period of
time, then the initial conditions are considered to lead to a steady-state periodic gait.
Ideally, the area over which this convergence occurs should be large.
The results of two separate tests are documented. The first test evaluates a subspace of the initial height and forward velocity. The height was varied over the range
0.3 m < z < 0.8 m in 0.1 m increments. Initial test contact was adjusted appropriately.
The velocity was varied over the range 0.2 m/sec < i < 2.0 m/sec in 0.1 m/sec increments. In Figure 4-9, marked areas represent initial conditions from which stable
gaits occur for a duration of at least 20.0 seconds. In all sampled cases, either the
simulation converged to a periodic cycle or instability occurred within the first 5.0
seconds.
Robustness to Initial Conditions
1.1
0.9
0.8
0.70.6
0.5-
0.4
0.30.2
.
0.2
0.4
0.6
0.8
1.4
1
1.2
Forward Velocity (nVsec)
1.6
1.8
2
Figure 4-9: Robustness to initial height and velocity: dark areas represent conditions
converging to a stable gait with duration of at least 20 seconds.
In the second test, the pitch and pitch derivative are varied to test orientation
robustness. Figure 4-10 displays initial conditions which result in stable runs. The test
samples initial pitch over the range -1.3 rad < 0 < 1.1 rad, in 0.01 rad increments,
and the initial pitch derivative 0 over the range -1.0 rad/sec < 6 < 1.0 rad/sec, in
0.5 rad/sec increments. Gaits lasting at least 10.0 sec were denoted as stable.
Somewhat surprisingly, a number of unstable initial pitch and derivative points lie
38
3 7E EI.IE F
Robustness to Pitch Conditions
1111 I
1
S0
-1
-2
-3
inrir7
-1
-4--1
-0.5
0
Pitch (rad)
0.5
1
Figure 4-10: Robustness to initial pitch and pitch rate of change: dark areas represent
conditions converging to a stable gait with duration of at least 10 seconds.
inside regions surrounded by stable initial conditions. I cannot produce a satisfactory
explanation of these unstable points. It may suggest that certain body orientations at
the instant of ground contact are more susceptible to instability caused by non-zero
angular momentum values. Further investigation of these conditions is necessary.
39
4.4
Parameter Sensitivity
Parameter sensitivity testing measures the amount which parameter values may vary
before the controller ceases to produce a stable long-term running gait. Very precise
parameters indicate that using the algorithm on a physical platform will require
very precise replication of the simulation and would therefore likely malfunction.
Parameter sensitivity also hints at which values have the greatest effect on the stability
of the system.
Parameter
kieg
Minimum
Nominal
675
750
Maximum
1100
13.5
6.0
3.5
-0.475
-0.2
-0-150
1.75
1.50
1.10
target
khip protract
1.49
(-52%)
2.12
(-65%)
3.104
0.310
0.400
6.062
retract
0.4
0.00
kg
0.550
rad
kg
0.94
rad
(+134%)
(-100%)
Table 4.3:
5.23
(+68%)
6.79
(+12%)
(+37.5%)
(-23%)
target #hip
m/sec
(+16%)
(-27%)
body mass
(leg mass constant)
total mass
(leg and body changed)
N-sec/m
(+137%)
(-25%)
Xmax
N-sec/m
(+125%)
(-42%)
Gve1 2
N/m
(+46%)
(-10%)
Gvel
Units
Independent sensitivity of parameter values:
minimum, nominal, and
maximum values for which a stable gait of more than 10.0 sec results.
A variation is made to an individual parameter value, and then the simulation
is tested from standard initial conditions. By varying only a single parameter and
holding the other values constant, the effects of that parameter can be isolated.
The parameter under test is reduced until a simulated trial fails to maintain a
stable gait for at least 10 seconds. The last value for which a stable gait occurred is
recorded. This lower bound is refined to a precision of at least two significant figures
by binary search. The test is repeated, increasing the parameter to establish an upper
bound. Results are tabulated in Table 4.3.
This test should only be interpreted as a "zeroth-order" evaluation of sensitivities.
Due to the nonlinearity of the system, the results are specific to the operating point
and conclusions cannot be made about the simultaneous variation of multiple parameters. Furthermore, the results cannot guarantee all parameter values lying within
the minima and maxima produce stable gaits. There exists the possibility that some
"island of instability" lies between sampled points. Although only anecdotal, it is
encouraging to note that no such regions were discovered during testing.
40
The parameters which permitted the smallest variation were kie, and the total
system mass; both parameters changed by around 10%, but in opposite directions.
One possible speculation supported by this is decreasing kie, and increasing system
mass may produce similar effects on motion in the controller. Body mass could be
increased by over 2 kg before instability. Since leg masses are held constant, the overall
system mass is increased by 2 kg and the proportion of body-to-leg mass changes. But
increase of total mass (while maintaining body-to-leg proportions) causes instability
after only 0.73 kg. This argues that the effect of leg mass on stability is more critical
than that of body mass.
Additionally, without changing any other parameter value, speeds may be targeted
over a 0.65 m/sec range without effecting stability. This single parameter speed
selection was impossible in the quadruped trotting model; changing target speed
required re-tuning a number of other parameters [13]. Finally, the insensitivity to
the target leg protraction angle Ostance protract is surprising in that previous control
approaches have required a high degree of precision to properly regulate a trade-off
of velocity and apex height [31].
41
4.5
Biological Plausibility
How can the similarity of biological running gaits and the gait generated here be
quantitatively evaluated? Since the morphology of the system models no specific
species, a direct comparison is impossible. Instead, cross-species metrics from the
literature are calculated for the simulated hexapod gait. These metrics are then
compared to values predicted by two biological models. Table 4.4 tabulates these
calculations and predictions along with the size of one standard deviation variance.
The table also includes brief descriptions of the metrics and their sources.
Gait
Metric
Sim
Bio
Fgrf
165
172.9
Al
0.042
0.073
00
0.40
0.56
to
0.13
0.089
0.139
Bio 1 Stdev
Units
Description
/ -38.54
+0.023 / -0.017
+0.10 / -0.86
N
peak vertical GRF
[10]
meters
peak leg compression
[10]
rad
leg sweep half-angle
[10]
sec
ground contact time
[10]
0.086
/ -0.014
+0.008 / -0.007
sec
half resonant period
[10]
1.18
4.63
+1.44 / -1.10
Watts/kg
leg-spring work rate
[10]
kleg
3.956
2.39
+0.74 / -0.57
kN/m
combined leg stiffness
[10]
kvert
7.86
1.78
+1.56 / -1.31
±0.37
kN/m
-
equivalent vertical stiffness
unitless peak force measure
[10]
F9 ,.
7.50
2.77
Fgrf leg- 1
0.92
0.78
±0.29
-
unitless peak force per leg
[4]
0.168
0.096
±0.02
-
unitless leg comression
[4]
krel
16.48
18.60
±1.68
-
relative stiffness
[4]
krelleg- 1
5.48
7.82
±2.26
-
relative stiffness per leg
[4]
T
2
Pspring
+49.61
+0.016
Source
[4]
Table 4.4: Comparison of simulation measurements with biological model predictions.
These numerical predictions take one of two forms. Either they are logarithmic fits
of observed data to functions of organism mass, or they are dimensionless quantities
calculated from polypedal trotters. In both cases, they derive from simplified springmass models of running, and most are measured across multiple morphologies, gaits,
and species. Limitations of space prevent the in-depth discussion of the meaning of
the metrics, the assumptions of the models from which they derive, or the procedure
by which biological values were obtained; instead, the reader is directed to the cited
literature.
The peak ground reaction forces (Fgf), ground contact times (t,), half-resonant
1
period (T/2), vertical stiffness (kvert), and normalized force per leg (Fgrjleg- ) all
lie within a standard deviation of biological predictions. A possible conclusion: the
"aerial to stance to aerial" bouncing behavior of the system is highly similar to bio42
logical systems. All metrics related to the stiffness of the leg lie within two standard
deviations, indicating that the system stiffness may only partially replicate that of
biological systems.
It is worth noting the value furthest afield, Ppring, is not unexpected. The metric
measures the energetic equivalence of the leg behavior with a passive linear spring.
Since the model of the leg the hexapod is just that, a value near 1.0 is expected.
Natural systems do not manage quite this level of mechanical efficiency. The deviation
from biological is a consequence of the morphological model.
43
44
Chapter 5
Conclusions and Future Work
This work makes the following contributions:
1. A new hexapedal locomotion controller which demonstrates a stable and fast
gait in simulation.
2. Results indicating this gait responds favorably to variations in initial conditions
and momenta during dynamic locomotion.
3. New numerical evidence suggesting that the class of controllers originating in
the quadrupedal trotting model exhibit robust locomotion in hexapedal morphologies.
Although the simulated control algorithm has evaluated favorably, at least three
far-reaching challenges must be met before the system described here reaches its full
potential.
First, tests on a physical system must be performed. Without this confirmation,
there is always the possibility some real-world factor was overlooked in these simulations. Currently, planned testing will be performed on the Scorpion robot. Should
this fall through, or turn out mechanically implausible, physical trials will require
the design and production of a new robot. In either case, such an evaluation is a
necessary prerequisite to acceptance of the work here.
Second, the tedious hand-tuning procedures need to be automated in some fashion. Hand-tuning requires significant effort from an experienced individual with intuition for parameter choices. Additionally, many of the hypotheses made about this
algorithm take the form, "This algorithm can produce some behavior." The handtuning procedure can never disprove such a statement, it can only prove by example.
There always exist more unchecked combinations of parameters. Hand-tuning's laborintensity and inability to negate certain conclusions make its automation a priority.
Finally, a more fundamental understanding of this algorithm needs to be developed. How do the control parameters relate to the morphology? What additional
high-level feedback control could enhance stability? Can the algorithm be further simplified? What biological conclusions can be confirmed? A more advanced theoretical
45
framework is necessary to address these questions. Mechanisms such as leg retraction, angular momentum matching, and ground contact prediction need additional
theoretical explanation.
I hope the simple algorithm presented in this thesis, one of only a documented
few capable of producing dynamically-stable hexapedal running gaits, will encourage
and aid further legged locomotion research.
46
Appendix A
Mechanical Designs
This appendix briefly documents three mechanical designs I completed. The parts
detailed here attempt to retrofit the Scorpion, a statically-stable walking hexapod
robot, to enable it to operate with the algorithm detailed above. These segments
replace the most distal segment of the legs. They increase the stroke length and
stiffness to the values required by the simulation.
The initial passive distal segment design provides a stroke length of two inches. An
interchangeable helical compression spring sets the stance stiffness, and a linear-stroke
potentiometer measures segment compression. Figure A-i shows CAD drawings of
the complete distal assembly.
I
II1
-0---------
Figure A-1: Assembly drawings and section view of first distal segment design.
Due to external deadlines, time pressure dictated a simple design. Parts were
47
conceived to be produced by a novice machinist with only a week of machining time;
nine segments were produced in seven business days by a commercial shop.
The unit is housed in an aluminum tube casing chosen from stock tubing sizes and
turned down to a requisite thickness. Two interior rings affix to the inner housing,
oriented coaxially, with a small gap between them. These rings hold a polymer thrust
bushing. A smaller tube moves in and out of the housing along this bushing. A motion
stop, consisting of a washer pressfit onto the slide, holds the slide inside the housing
during extension. An additional light compression spring provides return force for
the potentiometer.
After production, trials found that the overall weight of the initial design may be
too high. Plans were also altered, focusing on the eight-legged model of the Scorpion.
This necessitated manufacture of more segments, so I took the opportunity to tune
the design.
Figure A-2 shows the revised passive distal segment design. Its function is essentially the same as the initial design. The revision's improvements include:
1. Replacement of exterior screws. These screws may bind on the environment,
over-constrain the design, and add a disassembly risk. The screws were replaced
with interior threads.
2. Unnecessary metal was removed, reducing weight.
3. Reduced part count and assembly.
4. The bearing holder was eliminated, and instead the bearing is secured directly
to the outer distal housing. This reduces weight and part count, rotationally
confines the bearing, and presents easier assembly.
5. The washer pressfit to the top of the lower slide was removed. This washer acts
as a motion stop. There was concern that the pressfit glue would fail, allowing
the slide to separate from the distal segment. It was replaced with a machined
lip integral to the slide.
6. Grooves to hold the spring in position were added.
7. A metal sleeve on the potentiometer holder protects the potentiometer from
buckling due to misalignment.
8. The interface screw was too long, causing a visible gap between the hip and the
distal segment.
Finally, additional work went into a preliminary pneumatic design (Figure A-3).
The appeal of this design was its ability to inject energy during stance. Questions
had arisen as to the Scorpion's eventual ability to provide the hip power required in
the running algorithm. The initial power and energy analyses indicated that a small
air tank could provide sufficient energy to operate the legs for the duration of battery
life. The design was eventually scrapped for non-technical reasons.
48
Figure A-2: Assembly drawing and section view of revised distal segment
Figure A-3: Sectioned model of preliminary pneumatic distal segment
49
50
Appendix B
Control Source Code
The source code contained in this appendix:
8legs.h (Section B.1) is the header file which defines morphological values.
create-hexahop.c (Section B.2) contains the code which generates the hexapod
morphology.
bug-hop.h (Section B.3) is the header file for control code.
bug-hop.c (Section B.4) is the code of the control presented in this thesis
Miscellaneous Functions (Section B.5) lists utility functions necessary for a complete understanding of the listed code.
I reduced the code to contain only functions directly relevant to the controller
presented in this thesis. Utility functions, data collection, and experimental code has
been removed. These files require Creature Library and SD-Fast in order to build the
simulation.
Table B.1 describes the identifiers used in the source below, along with correspondences with the variable names used previously in the thesis.
51
Table B.1: Variable names used in simulation source code and coresponding symbols
from text.
Physical State
ls.t
q.x
q.z
q.pitch
q.hipn-z
q.hipn-x
q.kneen
qd.x
qd.z
qd.pitch
On
qd.hipn-z
#n
qd.kneen
tau.hipn-z
tau.kneen
Controller State
Is.current-state
ls.next-state
Is. test-contact
gcifootn.fs
ls.tripod
Is.tangvel-d
Is.ttakeoff
Is.moi-n
Controller Parameters
in
ls.k-aerial-kn
ls.b-aerial-kn
Is.kLaeriallhip
Is.b-aerial-hip
ls.k-leg
Is.bknee
ls.maxvel
ls.hip-retract
ls.hip-extend
ls.knee-extend
ls.leg-ratio
t
x
z
0
in
simulation current time
body COM x position (in laboratory frame)
body COM z position (in laboratory frame)
body pitch angle
hip n protract/retract angle
hip n outward splay angle
knee n length of compression
body COM x velocity
body COM z velocity
body pitch rate of change
hip n protract/retract angular rate
knee n compression velocity
Tn
Fkn
test contact
In
Kaerial knee
Baerial knee
Kaerial hip
Baerial hip
kieg
bleg
Xmax
Oretract
#protract
lextend
1compress
ls.G-vel
Gvei
ls.Gvel2
continued on next page
Gvel2
FSM current state (AERIAL or STANCE)
FSM next state
estimated time of AERIAL phase end
foot n ground contact foot switch
current stance tripod
target leg retraction speed
time of STANCE phase end
moment of intertia of leg n about hip joint
AERIAL phase knee position gain
AERIAL phase knee position damping
AERIAL phase hip position gain
AERIAL phase hip position daming
front/rear linear stance leg spring stiffness
front/rear linear stance leg damping
maximum forward velocity
AERIAL phase hip retraction target position
AERIAL phase hip protraction target position
AERIAL phase knee extension length
swing knee retraction distance for STANCE
defined as 'extend - ls.leg-ratio
STANCE phase stance tripod hip velocity gain
STANCE phase swing tripod hip velocity gain
52
continued from previous page
Simulation Parameters
ls.control-dt
ls.plant-dt
ls.check-dt
ls.servo-dt
gnd-footn.k-x
gndifootn.bx
gnd-footn.k-z
gndlfootn.b-z
Servo Interface
ls.k-joint
Is.bjoint
ls.ff-joint
q-d.joint
ls.utau-joint
ls.ltau-joint
ls.k-stop
ls.b-stop
time increment to execute control code
step size for dynamic state evolution
time increment to check for dynamic drift
time increment to update servo control
ground model stiffness (x direction)
ground model viscosity (x direction)
ground model stiffness (z direction)
ground model viscosity (z direction)
joint position gain
joint position damping
joint feedforward torque
desired joint position
joint applied torque upper limit
joint applied torque lower limit
joint stop contact model stiffness
joint stop contact model viscosity
53
B.1
6legs.h
/* 6legs.h */
/* defined constants for hexahop body morphology */
/* Matt Malchano */
#define NUM-LEGS 6
/* number of legs */
#define BODYX 0.620
#define BODYY 0.145
#define BODY-Z 0.100
body length (sagittal plane) */
body width (frontal plane) */
body height (transverse plane) */
#define SHIN-Z 0.176
#define SHINM 0.0565
#define SHIN._RAD 0.024
shin length */
shin mass
shin cylinder and end-cap radius */
10
#define PIN-Z 0
#define PIN-M 0.0
/* pin link length (unused) */
/* thight link length (m) */
#define THIGH.Z 0.076
#define THIGH-M 0.0908 /* thigh link mass */
#define THIGH-RAD 0.15 /* thigh cylinder and end-cap radius */
#define BODY-M 3.104
20
/* body segment mass */
#define TOTALM (BODYM + NUMLEGS*(THIGHM + SHINM + PIN-M))
#define KNEE-MIN 0.0
#define KNEEMAX 0.2
/* knee joint stop, min compression */
/* knee joint stop, max compression */
#define HIPX-MIN -PI/2
#define HIPXMAX PI/2
/* hip joint stop, min angle */
/* hip joint stop, max angle */
#define HIPZMIN -PI/2
#define HIPZ-MAX PI/2
/* hip joint stop, min angle */
/* hip joint stop, max angle */
30
#define SPONGEZ 0.005 /* ground contact max penetration */
#define CONTACTZ (SHIN-Z+SHINRAD-SPONGEZ)
//the x positions of hip joint connections to the
// body indexed by hip joint number
// Oth ignored, since legs numbered from 1.
static const double PX[NUM-LEGS+1] = { 0.0,
0.5 * BODYX,
0.0,
-0.5 * BODYX,
-0.5 * BODYX,
0.0,
0.5 * BODYX };
40
// the y positions of hip joint connections to body
// indexed by hip joint number
// Oth ignored, since legs numbered from 1.
static const double PY[NUMLEGS+1] = { 0.0,
50
-BODY-Y, -BODY.Y, -BODY-Y,
BODY-Y, BODYY, BODY-Y, };
// the z positions of hip joint connections to
// indexed by hip joint number
// Oth ignored, since legs numbered from 1.
static const double PZ[NUMLEGS+1] = {
body
0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 };
60
54
B.2
createlhexahop.c
#include <stdio.h>
#include <cl-lib.h>
#include "8legs.h"
// function prototypes
void make-leg(int n);
void body-shape(;
void thigh-shape(int legno);
void shin-shape(int legno);
void pin-shapeo;
void subn(char*, int);
10
int
main(int argc, char* argv[])
{
int i = 0;
command-line(argc, argv);
begin-species("hexahop");
// confine the model to sagittal plane
new-link("base");
set-parent(GROUND);
new-joint("trainer");
set-planar-joint("x", "z", "pitch", 'x', 'z', 'y');
// this servo set limp.
servo(1, "pitch", PDFFSERVO, "ls.k.pitch", "is.b.pitch", "1s.ff.f.pitch");
20
new-link("body");
body-shape(;
30
}
//set-parent(FREE); // uncomment for 3D simulations
for(i = 1; i <= NUMLEGS; i++) make-leg(i);
end-specieso;
/* make-leg(int n) constructs the morphology of leg n.
" It perfoms some string replacement to generate the
" names for the required variables. It then calls all
* the CreatureLibraryroutines nececssary
40
void
make-leg(int n)
{
// various labels that appear in final sim application
char hip-z[12] = "hip#_z";
char hip-x[12] = "hip#_x";
char
char
char
char
char
k-hip-z[15] = "1s.khip#_z";
b-hip-z[15] = "1s.bhip#_z";
ff-hip.z[15]
"is.ffhip#_z";
utau-hip-z[20] = "1s.utauhip#_z";
ltau-hip-z[20] = "is.itauhip#_z";
char
char
char
char
k-hip-x[15] =
b-hip-x[15] =
ff-hip-x[15] =
utau-hip-x[20]
50
"1s.khip#_x";
"is.bhip#_x";
"1s.ffhip#_x";
= "is.utauhip#_x";
55
char
ltau-hip-x[20]
char
char
char
char
char
char
knee[12] = "knee#";
k-knee[15] = "is.k-knee#";
b-knee[15] = "1s.bknee#";
ff-knee[15] = "is.ff-knee#";
utau-knee[20] = "1s.utauknee#";
ltau-knee[20] = "1s.itauknee#";
char
char
char
char
shin[12] = "shin#";
foot[12] = "foot#";
thigh[12] = "thigh#";
pin[12] = "pin#";
= "1s.itau-hip#_x";
60
70
// replace '#' above with leg number
subn(hip-z, n); subn(hip-x, n); subn(k-hip-z, n); subn(b.hip-z, n); subn(ff-hip-z, n);
subn(k-hip-x, n); subn(b-hip-x, n); subn(ff-hip-x, n);
subn(knee, n); subn(k-knee, n); subn(b-knee, n); subn(ff-knee, n); subn(shin, n); subn(foot, n);
subn(thigh, n); subn(pin,n);
subn(utau-hip-z, n); subn(tau.hip-z, n); subn(utau-hip-x, n); subn(tau-hip-x, n);
subn(utau-knee, n); subn(ltau-knee, n);
// create the hip protraction/retraction joint
jointpin(hipz,'y');
limit(hip-z, HIPZMIN, HIPZ.MAX);
servo(i, hip-z, PD-FFSERVO, k.hip-z, b-hip-z, ff-hip-z);
user-servo(5, hip-z, "pd-ff.taulimit-servo", 5, k-hip-z, b-hip-z, ff-hip-z, utau-hip-z,
set-parent("body");
set-joint-offset(PX[n], PY[n], PZ[n]);
80
Itau-hip-z);
//
the pin link allows us to seperate the joint axes, but currently
//its length is 0.
newilink(pin);
pin-shapeo;
90
// create the hip sprawl joint
joint pin(hip-x, 'x');
limit(hip-x, HIPZ-MIN, HIPXMAX);
servo(i, hip-x, PDFFSERVO, k-hip-x, b-hip-x, ff-hip-x);
user-servo(5, hip-x, "pd-ff taulimit -servo", 5, k-hip-x, b-hip-x, ff-hip-x, utau-hip-x,
set-joint-offset(0.0,0.0,0.0);
Itau-hip-x);
100
// generate the thigh link and give it the appropriate shape
new-link(thigh);
thigh-shape(n);
// generate the knee joint
joint-slider(knee,'z');
limit(knee, KNEEMIN, KNEEMAX);
servo(i, knee, PDFF-SERVO, k-knee, b-knee, ff-knee);
user-servo(5, knee, "pd-ff taulimit-servo", 5, k-knee, b-knee, ff_knee, utau-knee,
set-joint-offset(0.0, 0.0, -THIGH-Z);
Itau-knee);
110
//
//
finally, generate the shin and create a ground contact test point
at the tip of the foot.
new-link(shin);
shin-shape(n);
ground-contact(foot, 0.0,0.0,-CONTACTZ);
I
/* subn(char* s, int n) destructively modifies string s,
* replacing all occurences of '#' with the single digit
* number n's character value.
120
void
subn(char* s, int n)
{
while(*s != '\0')
{
56
if(*s
s++;
}
'#)
*s = (char)(n+48);
}
/* body-shape() makes the body shape, a big rectangular box.
* It also adds ground contact detectors to the bottom corners.
130
void
body..shape()
{
begin-shape(;
use-density(0.25 * 1.76 * ALUMINUMDENSITY/4.0);
translate(0.0,0.0,-BODY-Z/2.0);
shape(SBRICK, BODY_X, BODY-Y, BODY-Z);
ground-.contact("body1
BODYTOT*0.35,-BODYY/2.0,-BODYZ/2.0);
ground..contact("body2", BODYTOT*0.35,BODYY/2.0,-BODYZ/2.0);
ground-contact("body3", -BODYTOT*0.35,BODYY/2.0,-BODY..Z/2.0);
ground-contact("body4", -BODY-TOT*0.35,-BODY-Y/2.0,-BODY-Z/2.0);
end-shapeo;
140
I
/* pinshape() generates a cylindrical pin which connects
* the two hip joints together.
150
void
pin-shape()
{
begin-shape(;
use-density(ALUMINUM-DENSITY/4.0);
translate(0.0,0.0, -THIGH..Z/100.0);
shape(SCYLINDER, THIGHZ/50.0,THIGH-RAD);
end-shapeo;
}
160
/* thigh-shape(int legno) creates the thigh shape. It
* is optionally passed a legno to customize the leg's
* mass (unused).
void
thigh-shape(int legno)
{
begin-shape(;
if((legno == 2 1| legno == 5) && DBLMIDMASS) {
use-density(2.0 * 0.25 * ALUMINUM-DENSITY/8.0);
} else use.density(0.25 * ALUMINUMDENSITY/8.0);
170
shape(SSPHERE,JOINT-RAD);
translate(0.0,0.0,-THIGH-Z);
shape(SCYLINDER, THIGH-Z,THIGHRAD);
shape(SSPHERE,JOINTRAD);
end-shape(;
180
/* shin-shape(int legno) creates the shin shape. */
void
shin-shape(int legno)
{
begin-shape(;
if((legno == 2 11 legno == 5) && DBLMID-MASS)
use-density(2.0 * 0.25 * ALUMINUM-DENSITY/8.0);
else
use-density(0.25 * ALUMINUMDENSITY/8.0);
translate(0.0,0.0, -SHINZ);
shape(SCYLINDER, SHIN_ZSHINRAD);
57
190
identity(;
rotate('y', 180.0);
translate(0.0,0.0,-SHINZ);
shape(SHEMISPHERE,SHIN-RAD);
end-shapeo;
}
200
B.3
bug-hop.h
#include
#include
#include
#include
<stdio.h>
<math.h>
"creature.h"
"81egs.h"
#ifndef __BUGHOPH
#define __BUG-HOP-H
#define AERIAL
*define STANCE
00
20
#define LEG.LEN
#define MTHIGH
#define MSHIN
(THIGH-Z + PINZ+ SHIN-Z)
0.46411
0.0293018
10
void bug-hop2();
static
static
static
static
void aerial-controller();
void stance..controller();
void angmo-instrumento;
double angmo(int, double, double, double, double, double*, double*, double*);
20
#endif
B.4
bugihop.c
#include "bug-hop.h"
/* bug-hop2() is the entry function into the control code.
* It is called once every control-dt seconds, examines the
" dynamic state of the robot, and sets the relevent servo
* variables and tau torque output variables to generate the
" gait. Depending on the FSM state, it calls either an aerial
* control function or stance control function.
void bug-hop2()
10
{
= ls.next-state;
switch((int)ls.current-state) {
case(AERIAL):
aerial controllero; break;
case(STANCE):
stance-controller(); break;
ls.current-state
}
/* aerial-control() implements AERIAL phase control */
void aerial-controller() {
58
20
ls.k-aerialkn;
ls.k-kneel
Is.b-kneel
= ls.b.aerial-kn * sqrt(Is.k-kneel * (THIGHM + SHINM));
ls.k-knee2
ls.b-knee2
= ls.k-knee3 =
= ls.b-knee3 =
ls.k-knee4
ls.b-knee4
= ls.k-knee5 = ls.k-knee6 =s.k.kneel;
= ls.b-knee5 = ls.b-knee6
ls.b-kneel;
ls.k.hiplz
= ls.k-aerial-hip;
ls.b-hipl-z = ls.b-aerial-hip;
ls.k-hip2-z
ls.b-hip2-z
= ls.k-hip3_z = ls.k-hip4-z = ls.k-hip5_z = ls.k-hip6_z = ls.k-hipl-z;
= ls.b-hip3_z = ls.b-hip4_z = ls.b-hip5-z = ls.b-hip6_z = ls.b-hipl-z;
30
if(ls.tripod == 0) {
q-d.hiplz = q-d.hip5_z = q-d.hip3-z = ls.hip-retract;
q-d.hip2-z = q-d.hip4_z = q-d.hip6-z = ls.hip-extend;
q-d.kneel = q-d.knee5 = q-d.knee3 = ls.knee-extend + ls.leg.ratio;
q-d.knee2 = q-d.knee4 = q-d.knee6 = ls.kneeextend;
}
else
{
q-d.hip2-z = q-d.hip4_z = q-d.hip6-z = ls.hip-retract;
q-d.hiplz = q-d.hip3-z = q-d.hip5_z = ls.hip-extend;
}
q-d.knee2 = q-d.knee4 = q-d.knee6 = ls.leg.ratio + ls.kneeextend;
q-d.kneel = q-d.knee3 = q-d.knee5 = ls.knee-extend;
if(ls.t + ls.control-dt >= ls.t.est-contact) {
// The controller will enter STANCE phase next
// control execution. Prepare for the transition.
if(ls.tripod
ls.k-kneel
=
ls.k-knee3
ls.k-knee5
1) {
ls.kleg;
ls.k-kneel
ls.b-knee5
+ ls.k-knee3;
= ls.b-knee3 = Is.b-knee;
ls.k-knee6 = ls.k-knee4 =
ls.b-knee6 = Is.b-knee4 =
else
50
=s.rkileg-mult * ls.kleg;
ls.b-kneel
}
40
ls.k-knee2
ls.b-knee2
= 0.6 * ls.k.leg;
= 0.3 * ls.b-knee;
{
ls.k-knee6
60
= Is.k-leg;
ls.k-knee4 = ls.rk-leg.mult * ls.kileg;
ls.k-knee2 = ls.k-knee4 + ls.k-knee6;
Is.b-knee2 = ls.b-knee4 = Is.b-knee6 = ls.b-knee;
Is.k-kneel = ls.k.knee5 = ls.k-knee3 = 0.6 *
Is.b-kneel = ls.b-knee5 = ls.b-knee3 = 0.3 *
ls.kileg;
ls.b-knee;
I
ls.k.hiph-z
=
ls.k~hip3_z
=
ls.k.hip5..z
=0.0;
70
Is.k-hip2..z
= ls.k-hip4_z = Is.k-hip6_z = 0.0;
Is.b..hipl-z = ls.b-hip3_z = ls.b-hip5_z = 0.0;
ls.b-hip2-z = Is.b-hip4-z = ls.b-hip6-z = 0.0;
ls.tangvel-d
(ls.tripod?ls.maxvel:-ls.maxvel);
Is.next-state
STANCE;
}
80
/* stance-controller() implements STANCE phase
oid stance-controller() {
// depending on the active tripod...
if(ls.tripod == 1) {
// stance leg velocity control loops
ls.ff-hipl-z = -ls.G-vel * (qd.hipl.z * (0.25
Is.ff-hip5_z = -ls.G-vel * (qd.hip5_z * (0.25
ls.ff-hip3_z = -ls.G-vel * (qd.hip3-z * (0.25
control */v
- q.kneel) - Is.tangvel-d);
- q.knee5) - Is.tangvel-d);
- q.knee3) - Is.tangvel-d);
59
90
// swing leg velocity control loops
ls.ff-hip4_z = ls.G.vel2 * (qd.hip4_z + (ls.moi-3/ls.moi-4) * qd.hip3_z);
ls.ff-hip2_z = ls.G-vel2 * (qd.hip2-z + (ls.moi_5/ls.moi_2) * qd.hip5_z);
ls.ff-hip6_z = ls.G-vel2 * (qd.hip6-z + (ls.moi-l/ls.moi_6) * qd.hiplz);
}
else
{
// stance leg
Is.ff-hip4_z =
ls.ff-hip2_z =
ls.ff-hip6_z =
}
velocity control loops
-1s.G-vel * (qd.hip4_z * (0.25 - q.knee4) + ls.tangvel-d);
-ls.G-vel * (qd.hip2_z * (0.25 - q.knee2) + ls.tangvel-d);
-ls.G-vel * (qd.hip6.z * (0.25 - q.knee6) + ls.tangvel-d);
100
// swing leg velocity control loops
Is.ff-hipl-z = ls.G-vel2 * (qd.hipl-z + (ls.moi_6/ls.moi-1) * qd.hip6_z);
ls.ff-hip5-z = ls.Gvel2 * (qd.hip5_z + (ls.moiL2/ls.moi-5) * qd.hip2-z);
ls.ffhip3_z = ls.G-vel2 * (qd.hip3_z + (ls.moi-4/ls.moi_3) * qd.hip4-z);
if(gc-footl.fs I gc-foot2.fs | gc-foot3.fs I gc-foot4.fs |I
// a ground contact has occurred during stance phase.
// this prevents leaving stance phase prematurely.
ls.gnd-contact = 1.0;
(&.110
gc.foot5.fs || gc-foot6.fs)
{
if( !gcfootl.fs && !gc.foot2.fs && !gc.foot3.fs
&&!gc-foot4.fs && !gc-foot5.fs && !gc-foot6.fs
&& ls.gnd-contact == 1.0) {
// All the feet have left the ground after a ground
// contact had occured, go ahead and switch into aerial phase.
ls.next-state = AERIAL;
// swap the swing/stance tripods
ls.tripod = !(int)ls.tripod;
ls.gnd-contact = 0.0;
120
ls.t.takeoff = ls.t;
// predict next ground contact.
Is.t-estcontact = ls.G.delay*(qd.z/9.812) + ls.ttakeoff;
}
/* angmo.instrument() calculates the moments of intertia for
* each leg about the hip axis.
void angmo-instrument()
ls.moi1 = 9.5545e-4 +
ls.moi_2 = 9.5545e-4 +
ls.moi-3 = 9.5545e-4 +
ls.moi_4 = 9.5545e-4 +
ls.moi_5 = 9.5545e-4 +
ls.moi_6 = 9.5545e-4 +
130
{
7.7044e-5
7.7044e-5
7.7044e-5
7.7044e-5
7.7044e-5
7.7044e-5
+
+
+
+
+
+
0.0293*sq(0.25
0.0293*sq(0.25
0.0293*sq(0.25
0.0293*sq(0.25
0.0293*sq(0.25
0.0293*sq(0.25
- q.kneel);
- q.knee2);
- q.knee3);
- q.knee4);
- q.knee5);
- q.knee6);
I
/* calcuate leg angular momentum in system center of mass frame.
* currently unused.
140
double angmo(int leg, double theta, double dtheta, double lknee, double dknee,
double* amx, double* amy, double*amz)
{
double
double
double
double
thighIcm, shinIcm;
s-mrxr-x, s.mrxr-y, s-mrxr-z;
t-mrxr-x, tLmrxr-y, t-mrxr-z;
1;
150
//
location of shin com
I = THIGHZ + (SHIN-Z/2) - lknee;
// revolution term moments about com.
thigh-Icm = (1/12)*M-THIGH*(sq(THIGHZ) + sq(THIGH-X)) + M-THIGH*(THIGH.Z/2)*(THIGHZ/2);
shinIcm = (1/12)*MSHIN*(sq(SHIN-Z) + sq(SHIN-X)) + M-SHIN*l*1;
60
// M(r X rdot): orbital term for shin and thigh
s-mrxr-x = M.SHIN*(-PY[leg]*cos(theta)*(-dknee) + PY[leg]*l*sin(theta)*dtheta);
s-mrxr-y = MSHIN*(PX[leg]*cos(theta)*(-dknee) - (sq(l) + PX[leg]*l*abs(sin(theta)))*dtheta);
s.mrxr-z = M-SHIN*(-PY[leg]*sin(theta)*(-dknee) + PY[leg]*l*cos(theta)*dtheta);
160
tLmrxr-x = M-THIGH*(-PY[leg]*(THIGH-Z/2)*sin(theta))*dtheta;
t-mrxr-y = MTHIGH*(sq(THIGH-Z/2) + PX[leg]*(THIGH-Z/2)*abs(sin(theta)))*dtheta;
t-mrxr-z = MTHIGH*(PY[leg]*(THIGHZ/2)*cos(theta))*dtheta;
}
// angular momentum totals for the leg.
*amx = s-mrxrx + t-mrxr_x;
*amy = thighIcm*dtheta + shin-Icm*dtheta;
*amz = s-mrxry + t-mrxr-y + thighIcm*dtheta + shin-Icm*dtheta;
//*amz = s.mrxr-z + tLmrxrz;
B.5
170
Miscellaneous Functions
These utility functions are excerpted from unlisted source files.
/* do-pd-ff-servo implements a propotional-dervative-feedforward
* torque servo. It is never directly called by the control code
* but instead is called every servo-dt, where it examines the
* Is state variables q, qd, q-d... and modifies the value of
* tau as appropriate.
void
do-pd-ffJservo(tau, q, qd, q-d, k, b,
double *tau, q, qd, q-d;
double k, b, ff;
{
*tau = -k*(q
}
ff)
10
- q-d) - b*qd + ff;
/* nonlinear-spring-damper-groundhandles ground contacts.
* A PD feedback loop, along with the gnd-contact stiffness
* and damping values generates unidirectionalforces which
* prevent slipping and pentration.
void
nonlinear-spring-damperground(gc, x, y, z, dx, dy, dz)
basic.gcontact *gc;
double x, y, z, dx, dy, dz;
{
20
if (z > 0.0)
gc->fs
=
0.0;
if (z < 0.0)
if (gc->fs < 0.5)
{
gc->fs =
gc->td-x
gc->td-y
gc->tdz
if (gc->fs
{
1.0;
= x;
= y;
= z; /* Note that td-z isn't actually used, zero is */
30
> 0.5)
/* Foot switch is on, need to apply forces. */
/* x direction */
gc->f-x = (gc->td-x - x);
if (x > gc->td-x) /* f.x is positive, the force should be negative */
gc->Lx = -(gc->k-x)*gc->fLx*gc->f-x
- gc->b-x*dx;
else
61
40
gc->Lx = (gc->kx)*gc->f-x*gc->f-x
- gc->b-x*dx;
/* y direction */
gc->f-y = (gc->td-y - y);
if (y > gc->td-y) /* f-y is positive, the force should be negative */
gc->Ly = -(gc->k-y)*gc->fLy*gc->f-y
- gc->b-y*dy;
else
gc->f-y = (gc->k-y)*gc->f-y*gc->fLy
- gc->b-y*dy;
z direction */
gc->Lz = (gc->td-z - z);
/* You know z < gc->td-z or you wouldn't be in this part of the code */
gc->f.z = (gc->k-z)*gc->Lz*gc->f-z
- gc->b-z*dz;
/* Zero z direction if it would result in sticky ground. */
if (gc->fLz < 0.0) gc->f-z = 0.0;
50
/*
60
gc->t-th = 0.0;
}
else
{
gc->Lx = 0.0;
gc->Ly = 0.0;
gc->Lz = 0.0;
gc->tth = 0.0;
70
}
}
/* pdf-taulimit-servo adds output saturation to
* the standard proportional-derivative-feedforward
* servo model
void
pd-ffAtaulimit-servo(tau, q, qd, q-d, k, b, ff, utau,
double *tau, q, qd, q-d;
double k, b, ff;
double utau, Itau;
{
80
ltau)
*tau = -k*(q - q.d) - b*qd + ff;
if(utau > 0.0 && *tau > utau) *tau = utau;
ltau;
if(ltau < 0.0 && *tau < ltau) *tau
}
90
/* some math functions */
double
sq(double a) {
return a * a;
}
double
dmin(double a, double b)
if(a < b) return a;
else return b;
{
100
}
double
dmax(double a, double b)
if(a >= b) return a;
else return b;
{
}
62
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